Abstract
In this article we study how symmetries (invariance properties with respect to appropriate group actions) of periodic and quasiperiodic functions on \(\mathbb R^n (n \in \mathbb N)\) manifest themselves as patterns in Fourier space, i.e. as specific relations between certain Fourier coefficients of such a function. This is motivated by the experimental method of X-ray diffraction in crystallography, by which the atomic structure within a crystal can be determined. Mathematically, this tool heavily relies on Fourier analysis. In fact, it (approximately) produces the Fourier expansion of the corresponding electron density distribution function. Our results confirm that especially certain symmetries of this function are detected in that way. The case of quasiperiodic functions is related to quasicrystals.
Dedicated to Bernold Fiedler on the occasion of his 60th birthday.
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References
Chalmers, M.: A structural revolution. ESRF news 66, 16–17 (2014)
Duneau, M., Katz, A.: Quasiperiodic patterns. Phys. Rev. Lett. 54(25), 2688–2691 (1985)
Elser, V.: The diffraction pattern of projected structures. Acta Cryst. A 42, 36–43 (1986)
Gardner, M.: Extraordinary non-periodic tiling that enriches the theory of tiles. Sci. Am. 236, 110–121 (1977)
Guinier, A.: X-Ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies. Dover, New York (1994)
International Union of Crystallography. Report of the Executive Committee for 1991, Acta Cryst. A 48, 922–946 (1992)
Kramer, P., Neri, R.: On periodic and non-periodic space fillings of \(E^m\) obtained by projection. Acta Cryst. A 40, 580–587 (1984)
Lifshitz, R.: Introduction to Fourier-Space Crystallography. Balatonfüred (Hungary), Lecture Notes for the International School on Quasikristals (1995)
Mackay, A.L.: De nive quinquangula: On the pentagonal snowflake. Sov. Phys. Cryst. 26(5), 517–522 (1981)
Mackay, A.L.: Crystallography and the penrose pattern. Physica A 114, 609–613 (1982)
Penrose, R.: The role of aesthetic in pure and applied mathematical research. Bull. Inst. Math. and its Appl. 10, 266–271 (1974)
Rupp, F., Scheurle, J.: The role of Fourier analysis in X-ray crystallography. In: Hagen, Th, Rupp, F., Scheurle, J. (eds.) Dynamical Systems, Number Theory and Applications, 197–209. World Scientific, Singapore (2016)
Sands, D.E.: Introduction to Crystallography. Dover, New York (1993)
Shechtman, D., Blech, I.A., Gratias, D., Cahn, J.W.: Metallic Phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53(20), 1951–1953 (1984)
Urban, K., Kramer, P., Wilkens, M.: Quasikristalle. Phys. Bl. 42(11), 373–378 (1986)
Warren, B.E.: X-Ray Diffraction. Dover, New York (1969)
Acknowledgements
I would like to thank Florian Rupp from the German University of Technology in Oman (GUtech) for valuable discussions concerning the subject of this article. Also, I thank the referee for various helpful suggestions.
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Scheurle, J. (2017). Patterns in Fourier Space. In: Gurevich, P., Hell, J., Sandstede, B., Scheel, A. (eds) Patterns of Dynamics. PaDy 2016. Springer Proceedings in Mathematics & Statistics, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-64173-7_2
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DOI: https://doi.org/10.1007/978-3-319-64173-7_2
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